A Computational First Principle Examination of the Elastic, Optical, Structural and Electronic Properties of AlRF3 (R = N, P) Fluoroperovskites Compounds

This work describes an ab initio principle computational examination of the optical, structural, elastic, electronic and mechanical characteristics of aluminum-based compounds AlRF3 (R = N, P) halide-perovskites. For optimization purposes, we used the Birch–Murnaghan equation of state and discovered that the compounds AlNF3 and AlPF3 are both structurally stable. The IRelast software was used to compute elastic constants (ECs) of the elastic properties. The aforementioned compounds are stable mechanically. They exhibit strong resistance to plastic strain, possess ductile nature and anisotropic behavior and are scratch-resistant. The modified Becke–Johnson (Tb-mBJ) approximation was adopted to compute various physical properties, revealing that AlNF3 and AlPF3 are both metals in nature. From the density of states, the support of various electronic states in the band structures are explained. Other various optical characteristics have been calculated from the investigations of the band gap energy of the aforementioned compounds. These compounds absorb a significant amount of energy at high levels. At low energy levels, the compound AlNF3 is transparent to incoming photons, whereas the compound AlPF3 is somewhat opaque. The examination of the visual details led us to the deduction that the compounds AlNF3 and AlPF3 may be used in making ultraviolet devices based on high frequency. This computational effort is being made for the first time in order to investigate the aforementioned properties of these chemicals, which have yet to be confirmed experimentally.


Introduction
As the world's population is growing day by day, it is causing energy consumption to increase across the world. This results in a decrease in energy. For this purpose, new sources of energy, which are free from environmental problems, are necessary. Solar energy band gaps are the best options. These components may be used to produce complicated lattice-matched compounds with large band gaps, permitting lattice-matching and group space engineering [1]. The huge energy band gap is a property shared by several substances. Because of their great potential and low absorption edges, these compounds may be employed as glass in vacuum ultraviolet and ultraviolet wavelengths [9][10][11]. Recently, some research work on such compounds has been published, which may be referred to in the references section [12][13][14]. Harmel et al. [15] utilized DFT to investigate various properties of BaCsF 3 fluoro-perovskites, finding that CsBaF 3 may be useful for optoelectronic devices because of its wide direct bandwidth and groups of the unreal component of the shielding properties in the ultraviolet field. Daniel et al. [16] investigated the important properties of LiBaF 3 and discovered that these chemicals are more suitable for use in preparing devices for storing energy.
The new ternary family compound AlRF 3 (R = N, P) has the capacity to be used as a visual material in current electronic technologies. The composites that have forbidden a region width wider than 3.10 eV show better results in the ultraviolet spectrum. Because they are electrical conductors with great transparency across a small range of energies, AlNF 3 and AlPF 3 compounds exhibit metallic characteristics and are ideal candidates for electrical applications. The basic goal of the current research effort is to practice the DFT and the FP-LAPW approach for investigating the fundamental characteristics, such as the electronic, elastic, optical and structural properties of AlRF 3 (R = N, P) fluoro-perovskites, which may be used as core data for upcoming experimental research efforts on the materials listed above.

Computational Methodology
The full potential linearized augmented plane wave (FP-LAPW) technique was recycled to calculate the data for these substances using the coding computer application WIEN2K [17][18][19][20][21][22]. The electronic and other parameters, such as optical, density of state and so on, were calculated using the Tran-Blaha modified Becke-Johnson exchange potential approximation (TB-mBJ) technique. GGA approximation was used to calculate the exchange-correlation potential, but this method resulted in an incorrect band gap value and needed to be adjusted using either the TB-mBJ method or GGA plus the multi-orbital mean-field Hubbard potential (GGA + U). The optical characteristics and band gap calculated using these methods corresponded well with the experimental data [23,24]. In this research, we looked into the particular FP-LAPW technique with the smallest radii in muffin-tin spheres (RMT), which was equal to eight, and, in our case, K max , which was the norm of the most extreme K in the plane wave expansion. R = (N, P) elements and F had RMTs of 1.64, 1.77 and 2.06 au. The Fourier extended charge density was reduced to G max = 13.0 au in the muffin-tin region, while the spherical harmonics were extended to l max = 11.0. The self-consistent field values were observed to have been satisfied when the total energy was within the range of 0.001 Ry. Birch-Murnaghan's equation of state [25] allowed us to draw a graph of energy vs. volume, from which we extracted the structural variables. The elastic constants (ECs) of our chosen crystals were obtained using the computer program IRelast, and the compounds' elastic properties were then calculated [26]. The dielectric permeability was also recycled to assess the visual characteristics [27,28].

Results and Discussion
This segment of the study includes a full technical explanation of the results obtained utilizing the Tb-mBJ potential approaches.

Structural Properties
The unit cell of AlRF 3 (R = N and P) crystals with space group Pm3m (# 221) had a cubic perovskite structure with a single molecule. The F atoms appeared at ( 1 /2 , 1 /2, 1 /2); the R atoms (R = N, P) at (0, 1 /2, 1 /2), ( 1 /2, 0, 1 /2) and ( 1 /2, 1 /2, 0); and the Al atoms at (0, 0, 0), respectively. Figure 1 depicts the arrangement of Al-based fluoro-perovskites cubic crystals. We calculated the total energy of the cell against volume to be around V 0 (the volume of  [25], the volume optimization technique was used to obtain structural properties. We performed analytical calculations of our acquired locations using Birch-Murnaghan's equation of state to fit the obtained information and achieve the lowest energy state features, such as an equilibrium lattice length a o , a bulk modulus B and its pressure derivative B'. As illustrated in Figure 2, the bottommost energy unit cell could be obtained by reducing the total energy for the appropriate volume of the unit block. The lowest state energy E o was defined as the total minimum energy against volume. The volume was referred to as V o , which stands for the optimal least volume. The compounds constructed with the peak optimal energy were expected to be more stable. The ideal structural parameters which we calculated are included in Table 1, including a o (optimized lattice constant), E o (optimized lowest energy state), B (bulk modulus), V o (optimized volume) and B' (bulk modulus pressure derivative). Since the bulk modulus dropped with an increasing lattice constant, these consequences were consistent with the general trend of this approximation, implying that the estimated conclusions are very precise and genuine.

Electronic Properties
In this section, we determined the real diagrams of energy band structures and density of states (DOS) to explore the electronic characteristics of AlRF3 (R = N, P) crystals. The basic forbidden region of semiconductors and insulating materials is widely known to be

Electronic Properties
In this section, we determined the real diagrams of energy band structures and density of states (DOS) to explore the electronic characteristics of AlRF3 (R = N, P) crystals. The basic forbidden region of semiconductors and insulating materials is widely known to be

Electronic Properties
In this section, we determined the real diagrams of energy band structures and density of states (DOS) to explore the electronic characteristics of AlRF 3 (R = N, P) crystals. The basic forbidden region of semiconductors and insulating materials is widely known to be underestimated by local density approximations and GGA simulations [29,30]. The bulk of this is composite because its basic characteristics fail to produce both the exchange-correlation energy and its charge derivatives steadily. To solve band gap underestimation, the TB-mBJ was utilized, as it has been used effectively in recently published works [14,31,32]. Figure 3 illustrates the energy states obtained for AlMF 3 (M = N and P). The Fermi level energy was selected as the null-energy level at the topmost of the valence band (VB). Since the VB maximum and conduction band (CB) minimum values overlap for AlNF 3 and AlPF 3 , both are metals. Figure 4 depicts the total density of states (TDOS) and partial density of states (PDOS) for AlRF 3 (R = N, P) crystals, providing a detailed understanding of the electronic structure. DOS shows how numerous electronic positions contribute to both bands. The perpendicular dashed lines at zero electron volts represent the Fermi energy level E F , whereas the DOS ranges from −12.5 to 10 eV. The conduction band is the group of states, which is on the right-side of E F line, while the valance group is to the left. The F-tot, Al-tot, F-p and Al-s states in the VB have prominent contributions.

Elastic Properties
The elastic property is the reaction to external pressure by a crystal. This property is used to calculate the crystal's stability. The IRelast package, which was included in Wien2k and is specially built for cubic systems, was used in the calculation of ECs [33]. The three distinct ECs are C 11 , C 12 and C 44 , which are recycled to achieve ductility and mechanical stability in any particular crystal. Table 2 summarizes these independent constants. For a cubic crystal construction to be mechanically stable, the following ECs criteria should be satisfied: C 11 − C 12 > 0, C 11 > 0, C 44 > 0, C 11 − C 44 > 0, C 11 + 2C 12 > 0 and B > 0 [34]. The ECs and C ij values determined here reflect the compound's elastic stability. AlPF 3 has a C 11 value of 82.03 GPa, which is lower than AlNF 3 's value of 117.78 GPa. As a result, AlPF 3 is somewhat harder than AlNF 3 . The tendency of materials to create microscopic cracks is thoroughly connected to the crystal's elastic anisotropy, represented by A, which may be used specifically in the study of engineering. The value of A was obtained for the aforementioned compounds with the help of the obtained values of C 11 , C 12 and C 44 , according to Equation (1). Because the values of A for both the compounds were not equal to one, we were able to claim that our compounds possessed an anisotropic nature, and that the degree of anisotropy was indicated by the amount of variance. According to Table 2, this was 0.28 for AlPF 3 and 0.48 for AlNF 3 , which means that AlNF 3 is more anisotropic in nature. "Shear modulus G, Young's modulus E and Poisson ratio v" were calculated using ECs, with the help of the following formulas [35][36][37]: Table 2 shows the E, A, v and G values obtained from the equations above. A variety of factors can be recycled to assess the material's ductility or brittleness. (C 11 − C 44 ) is the compound's ductility, while Cauchy's pressure is the difference between C 11 and C 44 [38]. If C 11 − C 44 has a positive value, the compounds will be ductile, and if it is negative, the compounds will be brittle. Both materials have a positive Cauchy's pressure value, which is 0.49 GPa for AlPF 3 and 0.45 Gpa for AlNF 3 , showing that they are ductile. The Pugh ratio, or B/G, is the second methodology used for determining whether it is brittle or ductile. The greater value of "B/G" from the critical value of 1.75, the greater the compound's ductility will be and vice versa [39]. Our compounds were ductile because their values were 44.84 and 9.91 for AlPF 3 and AlNF 3 , respectively. This means that AlNF 3 is somewhat less ductile than AlPF 3 . In order to differentiate between the ductile and brittle nature of any compound, T. Frantsevich et al. [40,41], using v, obtained 0.26 as the critical value. Therefore, if a material's "v" value is less than 0.26, it will be brittle, and vice versa. Table 2 shows that both ternary AlRF 3 (R = N, P) compounds had a value greater than 0.26, namely, 0.49 for AlPF 3 and 0.460 for AlNF 3 , signifying its ductile nature. Finally, we discovered that the AlRF 3 (R = N, P) compounds are mechanically ductile, anisotropic, tough and fracture-resistant. Based on these results, we can definitely see the elastic properties being used in a range of current electrical devices.

Optical Properties
We subjected our materials to packets of light with energies ranging from 0 to 14.5 eV, and all the visual properties of our compounds were calculated using the predicted equilibrium lattice constant. The dielectric function ε(ω) was used to determine all optical characteristics.

The Refractive Index
Using ε 1 (ω) and ε 2 (ω), we determined several physical characteristics, such as the refractive index η(ω), optical conductivity σ(ω), absorption coefficient i(ω) and reflectivity r(ω) of our compounds. Figure 5 depicts the computed value of the refractive index 4.3 as 6, and the values of refractive index η(0) = 4.6 at 0 eV for the crystals AlPF 3 and AlNF 3 , as seen in the refraction spectrum. The curves of Figure 6 clarify that the refractive indexes of the aforementioned compounds slightly deviated from each other. Figure 5 shows that AlPF 3 had the highest refractive index value of 2.4 and 2.4 at photon energies of 5 and 10 eV, respectively, whereas the peak refractive index of AlNF 3 was 2.1 at 6.5 eV. We determined the refraction of light from a particular composite based on its refractive index, which was extremely useful for photoelectric purposes. The refractive index was more than 1 (η(ω) > 1) because when a photon enters a compound, it feels hindrance due to the photon-electron interaction, as revealed in the same Figure. The greater the ε(ω) of a substance when it passes through, the more photons are refracted. Each process that raises the electron density of a substance also enhances its refractive index.
Molecules 2023, 28, x FOR PEER REVIEW 8 of 12 of the aforementioned compounds slightly deviated from each other. Figure 5 shows that AlPF3 had the highest refractive index value of 2.4 and 2.4 at photon energies of 5 and 10 eV, respectively, whereas the peak refractive index of AlNF3 was 2.1 at 6.5 eV. We determined the refraction of light from a particular composite based on its refractive index, which was extremely useful for photoelectric purposes. The refractive index was more than 1 (η(ω) > 1) because when a photon enters a compound, it feels hindrance due to the photon-electron interaction, as revealed in the same Figure. The greater the ε(ω) of a substance when it passes through, the more photons are refracted. Each process that raises the electron density of a substance also enhances its refractive index.

The Absorption Coefficient
The absorption coefficient I(ω) was calculated using the dielectric function, as shown in Figure 6. From Figure 6, it is clear that our crystals have appreciable absorption coefficients at energies varying from 0 eV to 14 eV for AlPF3 and AlNF3. Both of these compounds have essentially identical critical values, which are 0.0 eV for AlPF3 and AlNF3, respectively. At energy levels of 0.50, 3.  Figure 7 explains the reflectivity R(ω), which was estimated from the dielectric permittivity exposed through the variation of energy from 0.0 eV to 14.0 eV. For AlPF3 and of the aforementioned compounds slightly deviated from each other. Figure 5 shows that AlPF3 had the highest refractive index value of 2.4 and 2.4 at photon energies of 5 and 10 eV, respectively, whereas the peak refractive index of AlNF3 was 2.1 at 6.5 eV. We determined the refraction of light from a particular composite based on its refractive index, which was extremely useful for photoelectric purposes. The refractive index was more than 1 (η(ω) > 1) because when a photon enters a compound, it feels hindrance due to the photon-electron interaction, as revealed in the same Figure. The greater the ε(ω) of a substance when it passes through, the more photons are refracted. Each process that raises the electron density of a substance also enhances its refractive index.

The Absorption Coefficient
The absorption coefficient I(ω) was calculated using the dielectric function, as shown in Figure 6. From Figure 6, it is clear that our crystals have appreciable absorption coefficients at energies varying from 0 eV to 14 eV for AlPF3 and AlNF3. Both of these compounds have essentially identical critical values, which are 0.0 eV for AlPF3 and AlNF3, respectively. At energy levels of 0.50, 3.  Figure 7 explains the reflectivity R(ω), which was estimated from the dielectric permittivity exposed through the variation of energy from 0.0 eV to 14.0 eV. For AlPF3 and

The Absorption Coefficient
The absorption coefficient I(ω) was calculated using the dielectric function, as shown in Figure 6. From Figure 6, it is clear that our crystals have appreciable absorption coefficients at energies varying from 0 eV to 14 eV for AlPF 3 and AlNF 3 . Both of these compounds have essentially identical critical values, which are 0.0 eV for AlPF 3 Figure 7 explains the reflectivity R(ω), which was estimated from the dielectric permittivity exposed through the variation of energy from 0.0 eV to 14.0 eV. For AlPF 3 and AlNF 3 , the reflectivity values at zero-frequency R(0) were 0.42 and 0.54, correspondingly. As the incident photon energy was enhanced, the reflectivity of both complexes first increased up to approximately 0.4 eV and then declined to 0.09 at 1.0 eV, and after 1.0 eV, it once Molecules 2023, 28, 3876 9 of 12 again rose. Afterward, the values separated, and AlNF 3 had values of 0.20, 0.35 and 0.4 at 5.2, 6.6 and 10.6 eV, respectively. Similarly, further peaks of 0.19, 0.45 and 0.4 were detected for AlPF 3 at 2.2, 60 and 10.5-0 eV, respectively. In comparison to AlNF 3 , AlPF 3 had an extremely low reflectance of 0 eV. After 1.0 eV, it stayed low for AlNF 3 compared to AlPF 3 at higher energies, indicating that AlNF 3 was more transparent than AlPF 3 in this energy range. The transparency of the material suggested that these crystals may be used to produce lenses.

The Reflectivity
AlNF3, the reflectivity values at zero-frequency R(0) were 0.42 and 0.54, correspondingly. As the incident photon energy was enhanced, the reflectivity of both complexes first increased up to approximately 0.4 eV and then declined to 0.09 at 1.0 eV, and after 1.0 eV, it once again rose. Afterward, the values separated, and AlNF3 had values of 0.20, 0.35 and o.4 at 5.2, 6.6 and 10.6 eV, respectively. Similarly, further peaks of 0.19, 0.45 and 0.4 were detected for AlPF3 at 2.2, 60 and 10.5-0 eV, respectively. In comparison to AlNF3, AlPF3 had an extremely low reflectance of 0 eV. After 1.0 eV, it stayed low for AlNF3 compared to AlPF3 at higher energies, indicating that AlNF3 was more transparent than AlPF3 in this energy range. The transparency of the material suggested that these crystals may be used to produce lenses.

Optical Conductivity
Photon conduction is mathematically represented as σ(ω), which describes the electrons' movement in a substance produced by the application of an electromagnetic field. We may study the conductivity σ(ω) using the dielectric function, as revealed in Figure 8. For both substances, photon conductivity began at 0 eV. Other peaks for AlPF3 were 1000, 0, 2200, 5800 and 2800 at 0.4, 2.0, 3.2, 5.9 and 10.3 eV, respectively. Figure 8 depicts the peaks for AlNF3 as 1000, 0, 3000, 3100, 1900 and 2000 at 0.4, 2.0, 5.0, 6.3, 9.9 and 11.5 eV, respectively. As a consequence, it was discovered that the AlPF3 crystal is optically more conductive at low energy than AlNF3.

Optical Conductivity
Photon conduction is mathematically represented as σ(ω), which describes the electrons' movement in a substance produced by the application of an electromagnetic field. We may study the conductivity σ(ω) using the dielectric function, as revealed in Figure 8. AlNF3, the reflectivity values at zero-frequency R(0) were 0.42 and 0.54, correspondingly. As the incident photon energy was enhanced, the reflectivity of both complexes first increased up to approximately 0.4 eV and then declined to 0.09 at 1.0 eV, and after 1.0 eV, it once again rose. Afterward, the values separated, and AlNF3 had values of 0.20, 0.35 and o.4 at 5.2, 6.6 and 10.6 eV, respectively. Similarly, further peaks of 0.19, 0.45 and 0.4 were detected for AlPF3 at 2.2, 60 and 10.5-0 eV, respectively. In comparison to AlNF3, AlPF3 had an extremely low reflectance of 0 eV. After 1.0 eV, it stayed low for AlNF3 compared to AlPF3 at higher energies, indicating that AlNF3 was more transparent than AlPF3 in this energy range. The transparency of the material suggested that these crystals may be used to produce lenses.

Conclusions
In the presented study, we effectively studied the structural, elastic, electronic and optical characteristics of ternary fluoro-perovskite AlRF 3 (R = N, P) crystals. The most accurate and recent results are as follows. Based on improved structural parameters, we determined that AlRF 3 (R = N, P) is cubic and structurally stable. The IRelast program was applied to guess flexible constraints, such as fundamental ECs, "anisotropy factor, Poison ratio, ductility, Cauchy's pressure, shear modulus, Pugh ratio and Young modulus". Based on these basic elastic properties, it was determined that the aforementioned crystals were elastically stable, anisotropic, scratch-resistant and ductile. As a result of these discoveries, we are confident in the use of these materials in a wide range of current electrical technologies. To investigate the basic electrical properties of our crystals, the TB-mBJ approximation was adopted. The chemicals AlPF 3 and AlNF 3 were revealed to be conductors. F-tot, Al-tot, F-p and Al-s states in the VB contributed the most to the DOS, with energies ranging from −12.0 to −2.50 eV, −4 to −2.50 eV, −12.0 to −2.50 eV and −7.5 to −2.5 eV for AlPF 3 . For AlPF 3 , the biggest contribution in the conduction band came from the Al-tot, F-tot, Al-p and P-p states, with energies ranging from 2.0 to 4.0 eV, −1.0 to 4.0 eV, 2.0 to 4.0 eV and −1.0 to 4.0 eV, respectively. In contrast, the greatest contributions to AlNF 3 were from F-tot, P-tot, F-p and N-p, from −8.0 to −2.50 eV, −8.0 to −2.50 eV, −10.0 to 0 eV and 10.0 to 0 eV, respectively. In the same compound's conduction band, the greatest contributions came from the Al-tot and Al-p states, with energies ranging from 2.50 eV to 10.0 eV and 2.5 to 8.0 eV, respectively.